ON A NONCOMMUTATIVE ALGEBRAIC GEOMETRY PIERRE DOLBEAULT Sorbonne Universit´es, UPMC Univ. Paris 06, Institut de Math´ematiques de Jussieu - Paris Rive Gauche UMR7586, 4, place Jussieu 75005 Paris, France E-mail: [email protected]

Abstract. Several sets of quaternionic functions are described and studied with respect to hyperholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamilton 4-manifolds analogous to Riemann surfaces, for H instead of C, are defined, and so begin to describe a class of four dimensional manifolds. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quaternions. H-valued functions. Hyperholomorphic functions . . . . . . . . . . . . . . 3. Almost everywhere hyperholomorphic functions whose inverses are almost everywhere hyperholomorphic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. On the spaces of hypermeromorphic functions . . . . . . . . . . . . . . . . . . . . . . . 5. Globalisation. Hamilton 4-manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Hamilton 4-manifold of a hypermeromorphic function . . . . . . . . . . . . . . . . . . 7. The Hamilton 4-manifold Y of F when X = HP . . . . . . . . . . . . . . . . . . . . . .

1 2 4 7 8 10 13

1. Introduction. We first recall the definition of the field H of quaternions using pairs of complex numbers and a modified Cauchy-Fueter operator (section 2) that have been introduced by C. Colombo and al., [CLSSS07]. We will only use right multiplication. We will consider C ∞ H-valued quaternionic functions defined on an open set U of H whose behavior mimics the behavior of holomorphic functions on an open set of C. If such a function does not vanish identically, it has an (algebraic) inverse. Finally we describe properties of Hyperholomorphic functions with respect to addition and multiplication. 2010 Mathematics Subject Classification: Primary 30G35; Secondary 30D30. Key words and phrases: quaternions, holomorphic, hypermeromorphic functions, Hamilton 4manifolds.

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In section 3, we characterize the quaternionic functions which are, almost everywhere, hyperholomorphic and whose inverses are hyperlomorphic almost everywhere, on U , as the solutions of a system of two non linear PDE. We find non trivial examples of a solution showing that the considered space of functions is significant; we will call these functions Hypermeromorphic. At the moment, I am unable to get the general solution of the the system of PDE. Same difficulty for subsequent occurring systems of PDE. In section 4, we describe a subspace of hyperholomorphic and hypermeromorphic functions defined almost everywhere on U , having “good properties for addition and multiplication”; we again obtain systems of non linear PDE. In section 5 and the following, we consider globalization of the above notions, define Hamilton 4-manifolds analogous to Riemann surfaces, for H instead of C, and give examples of such manifolds; our ultimate aim is to describe a class of 4-dimensional manifolds. 2. Quaternions. H-valued functions. Hyperholomorphic functions. See [CSSS04, CLSSS07, D13]. 2.1. Quaternions. If q ∈ H, then q = z1 + z2 j where z1 , z2 ∈ C. We have z1 j = jz 1 , and note |q| = |z1 |2 + |z2 |2 . The conjugate of q is q = z 1 − z2 j. Let us denote * the (right) multiplication in H, then the right inverse of q is: q −1 = |q|−1 q ∼ C2 and f ∈ C ∞ (U, H), 2.2. Quaternionic functions. Let U be an open set of H = then f = f1 + f2 j, where f1 , f2 ∈ C ∞ (U, C). The complex valued functions f1 , f2 will be called the components of f . 2.3. Definitions. Let U be an open neighborhood of 0 in H ∼ = C2 . (a) From now on, we will consider the quaternionic functions f = f1 + f2 j having the following properties: (i) When f1 and f2 are not holomorphic, the set Z(f1 ) ∩ Z(f2 ) is discrete on U ; (ii) for every q ∈ Z(f1 ) ∩ Z(f2 ), Jqα (.) denoting the jet of order α at q (see [M66]), let mi = supαi Jqαi (fi ) = 0; mi , i = 1, 2, is finite. mq = inf mi is the order of the zeroe q of f . (b) We will also consider the quaternionic functions defined almost everywhere on U (i.e. outside a locally finite set of C ∞ hypersurfaces, namely Z(f1 ), Z(f2 )). 2.4. Modified Cauchy-Fueter operator D. Hyperholomorphic functions. See [CLSSS07, F39]. For f ∈ C ∞ (U, H), with f = f1 + f2 j, 1 ∂ ∂
+j f (q). Df (q) = 2 ∂z 1 ∂z 2 A function f ∈ C ∞ (U, H) is said hyperholomorphic if Df = 0. Characterization of the hyperholomorphic function f on U : ∂f1 ∂f 2 ∂f1 ∂f 2 − = 0; + = 0, on U. ∂z 1 ∂z2 ∂z 2 ∂z1

(1)

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2.5. Several families of meromorphic functions. The conditions f1 is holomorphic and f2 is holomorphic are equivalent on U ; the same is true for almost everywhere defined holomorphic functions on U . By definition, holomorphic (almost everywhere defined functions of two complex variables on U are such that f2 = 0, and f1 is (almost everywhere) holomorphic. 2.5.1. Consider the almost everywhere defined hyperholomorphic functions on U whose components are real. f = f1 + f2 j According to a remark of Guy Roos in March 2013, they are almost everywhere holomorphic [R13]. 2.5.2. The above considered almost everywhere holomorphic functions are meromorphic and constitute two H-commutative algebras A1 , A2 , with common origin 0. Let f = a+bi, and g = c+dj, with a, b, c, d ∈ R be two almost everywhere defined holomorphic functions i.e.meromorphic functions on U . A1 is the set of the meromorphic functions f = a+bi, and A2 is the set of meromorphic functions g = c + dj, with a, b, c, d ∈ R The sums f + g = a + c + dj + bi constitute the algebra A1 + A2 of meromorphic functions. More generally, Aα,β = αA1 + βA2 , with α, β ∈ R is an algebra of meromorphic functions on UX . Aαβ = α(a + bi) + β(c + dj) a,b,c,d,α,β∈R

2.5.3. We now begin to introduce multiplication for hyperholomorphic functions, addition and scalar muliplication being obvious. 2.6. Multiplication of almost everywhere defined hyperholomorphic functions. Proposition 2.1. Let f 0 , f 00 be two almost everywhere defined hyperholomorphic functions. Then, their product f 0 ∗ f 00 satisfies: ∂ 0 ∂
00 D(f 0 ∗ f 00 ) = Df 0 ∗ jf 00 + f 0 ( )+f j f ∂z 1 ∂z 2 Proof. f 0 = f10 + f20 j, f 00 = f100 + f200 j be two hyperholomorphic functions. 00

00

We have: f 0 ∗ f 00 = (f10 + f20 j)(f100 + f200 j) = f10 f100 − f20 f 2 + (f10 f200 + f20 f 1 )j Compute ∂
0 00 1 ∂ 00 00
+j f1 f1 − f20 f 2 + (f10 f200 + f20 f 1 )j 2 ∂z 1 ∂z 2 By derivation of the first factors of the sum f 0 ∗ f 00 , we get the first term: 1 ∂f10 ∂f 0
1 ∂f20 ∂f 0
00 00 + j 1 (f100 + f200 j) + + j 2 jj(f 2 − f 1 j) 2 ∂z 1 ∂z 2 2 ∂z 1 ∂z 2 =

1 ∂f10 1 ∂f20 j ∂f 0
∂f 0 j
+ j 1 (f100 + f200 j) + + j 2 j(f200 j + f100 ) = Df 0 ∗ jf 00 2 ∂z 1 ∂z 2 2 ∂z 1 ∂z 2

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By derivation in
1 ∂ ∂
0 00 f1 f1 + f20 jf200 j + (f10 f200 j + f20 jf100 ) +j 2 ∂z 1 ∂z 2 of the second factors of the sum f 0 ∗ f 00 , we get the second term (up to factor 12 ):

f10

00 00 ∂f 00 ∂f100 0 ∂f 0 ∂f + f 1 j 1 + f10 2 j + f 1 j 2 j ∂z 1 ∂z 2 ∂z 1 ∂z 2 00 00 00 ∂f 00 ∂f 0 ∂f 0 ∂f + f20 j 2 j + f 2 j 2 + f20 j 1 + f 2 jj 1 ∂z 1 ∂z 2 ∂z 1 ∂z 2 ∂ ∂ 0 0 = (f10 + f20 j)( )(f 00 + f200 j) + (f 1 + f 2 j)j (f 00 + f200 j) ∂z 1 1 ∂z 2 1 ∂
00 ∂ 0 0 (f1 + f200 j) = (f10 + f20 j)( ) + (f 1 + f 2 j)j ∂z 1 ∂z 2 ∂ 0 ∂
00 = f 0( )+f j f . ∂z 1 ∂z 2

3. Almost everywhere hyperholomorphic functions whose inverses are almost everywhere hyperholomorphic. 3.1. Definitions. We call inverse of a quaternionic function f : q 7→ f (q), the function defined almost everywhere on U : q 7→ f (q)−1 ; then: f −1 = |f |−1 f , where f is the (quaternionic) conjugate of f , then: f −1 = |f |−1 (f 1 − f2 j). −1

−1

Behavior of f −1 at q ∈ Z(f ). Let n1 = supJqα (|f |f 1 ); n2 = supJqα (|f |f 2 ). Define : nq = sup ni , i = 1, 2 as the order of the pole q of f −1 . 3.2. Characterisation. Proposition 3.1. The following conditions are equivalent: (i) the function f and its right inverse are hyperholomorphic, when they are defined; (ii) we have the equations:

(f 1 − f1 )

f2

∂f 1 ∂f2 ∂f − f2 − f2 1 = 0 ∂z1 ∂z1 ∂z 2

∂f1 ∂f 2 ∂f + (f 1 − f1 ) − f2 2 = 0 ∂z1 ∂z1 ∂z 2

Proof. Let f = f1 +f2 j be a hyperholomorphic function and g = g1 +g2 j = |f |−1 (f 1 −f2 j)

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its inverse; so g1 = |f |−1 f 1 ; g2 = −|f |−1 f2 , where |f | = (f1 f 1 + f2 f 2 ). ∂
1 ∂g1 ∂g
1 ∂g1 ∂g
1 ∂ +j g(q) = − 2 (q) + j + 2 (q) Dg(q) = 2 ∂z 1 ∂z 2 2 ∂z 1 ∂z2 2 ∂z 2 ∂z1
∂g1 ∂f ∂f ∂f ∂f ∂f 1 2 f 1 + f1 1 + f 2 + f2 2 = |f |−1 1 − |f |−2 f 1 ∂z 1 ∂z 1 ∂z 1 ∂z 1 ∂z 1 ∂z 1
∂f ∂f ∂f ∂g 2 ∂f ∂f 1 2 f 1 + f1 1 + f 2 + f2 2 − = |f |−1 2 − |f |−2 f 2 ∂z2 ∂z2 ∂z2 ∂z2 ∂z2 ∂z2
∂g1 ∂f ∂f ∂f ∂f ∂f 1 2 = |f |−1 1 − |f |−2 f 1 f 1 + f1 1 + f 2 + f2 2 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂g 2 ∂f ∂f ∂f ∂f
∂f 1 2 f 1 + f1 1 + f 2 + f2 2 = −|f |−1 2 + |f |−2 f 2 ∂z1 ∂z1 ∂z1 ∂z1 ∂z1 ∂z1 2|f |2 Dg ∂f 1 ∂f1 ∂f2 ∂f ∂f ∂f 2 + ) − f 1 f1 1 − f 1 f 1 − f 1 f2 2 − f 1 f 2 ∂z 1 ∂z2 ∂z 1 ∂z 1 ∂z 1 ∂z 1 ∂f1 ∂f 1 ∂f2 ∂f 2 −f 1 f 2 − f1 f 2 − f 2f 2 − f2 f 2 ∂z2 ∂z2 ∂z2 ∂z2  ∂f 1 ∂f 2 ∂f1 ∂f2 ∂f ∂f +j (f1 f 1 + f2 f 2 )( − ) − f 1f 1 − f 1 f1 1 − f 1 f 2 − f 1 f2 2 ∂z 2 ∂z1 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂f1 ∂f 1 ∂f2 ∂f 2  +f 1 f 2 + f1 f 2 + f 2f 2 + f2 f 2 ∂z1 ∂z1 ∂z1 ∂z1

= (f1 f 1 + f2 f 2 )(

Use the fact: f is hyperholomorphic: ∂f1 ∂f 2 ∂f1 ∂f 2 − = 0; + =0 ∂z 1 ∂z2 ∂z 2 ∂z1

(1) 2|f |2 Dg =

∂f 2 ∂f ∂f1 ∂f ∂f1 ∂f2 ∂f2 + f2 f 2 1 − f 1 f 1 − f 1 f2 2 − f 1 f 2 − f 2f 2 + f2 (f1 − f 1 )+ ∂z2 ∂z 1 ∂z 1 ∂z 1 ∂z2 ∂z2 ∂z 1  ∂f ∂f2 ∂f ∂f1 ∂f1 ∂f ∂f2  +j +f2 f 2 1 −f 1 f 2 −f 1 f2 2 +f 1 (f1 −f 1 )+f 1 f 2 +f1 f 2 1 +f 2 f 2 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂z1 ∂z1 ∂z1 f1 f 1

f being hyperholomorphic, g hyperholomorphic is equivalent to the system of two equations:

+f1 f 1

∂f 2 ∂f ∂f1 ∂f ∂f1 ∂f2 ∂f2 + f2 f 2 1 − f 1 f 1 − f 1 f2 2 − f 1 f 2 − f 2f 2 +f2 (f1 − f 1 ) = 0 ∂z2 ∂z 1 ∂z 1 ∂z 1 ∂z2 ∂z2 ∂z 1

+f2 f 2

∂f 1 ∂f2 ∂f ∂f1 ∂f1 ∂f ∂f2 − f 1f 2 − f 1 f2 2 + f 1 (f1 − f 1 ) + f 1 f 2 + f1 f 2 1 + f 2 f 2 =0 ∂z 2 ∂z 2 ∂z 2 ∂z 2 ∂z1 ∂z1 ∂z1

f1 and f2 satisfy, by conjugation of the second equation: +f2 f 2

∂f1 ∂f2 ∂f2 ∂f ∂f ∂f ∂f − f1 f1 1 − f1 f 2 + f2 2 (f 1 − f1 ) + f1 f 1 − f1 f2 1 − f2 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂z 2

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∂f1 ∂f ∂f2 ∂f ∂f2 ∂f ∂f1 + f1 f 2 1 . + f 2 f 2 + f2 f 2 1 − f 1 f 2 − f 1 f2 2 + f 1 (f1 − f 1 ) = 0 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂z 2 ∂z 2 Using (1), we get: +f 1 f 2

+f2 f 2

∂f1 ∂f ∂f2 ∂f ∂f ∂f + f1 (f 1 − f1 ) 1 − f1 f 2 + f2 2 (f 1 − f1 ) − f1 f2 1 − f2 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2

∂f ∂f1 ∂f ∂f2 ∂f1 ∂f + (f1 − f 1 )f 2 1 + f 2 f 2 + f1 (f1 − f 1 ) + f2 f 2 1 . − f 1 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂z 2 Assume f1 6= 0, f2 6= 0

+f 1 f 2

∂f1 ∂f ∂f2 ∂f ∂f ∂f
+ f1 (f 1 − f1 ) 1 − f1 f 2 + f2 2 (f 1 − f1 ) − f1 f2 1 − f2 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂f1 ∂f ∂f2 ∂f ∂f ∂f
−f2 +f 1 f 2 +(f1 −f 1 )f 2 1 +f 2 f 2 −f 1 2 (f1 −f 1 )+f2 f 2 1 −f 1 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 By sum: ∂f ∂f2 ∂f
∂f f 1 f1 (f 1 − f1 ) 1 − f1 f 2 + f2 2 (f 1 − f1 ) − f1 f2 1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂f ∂f2 ∂f ∂f
−f2 (f1 − f 1 )f 2 1 + f 2 f 2 − f 1 2 (f1 − f 1 ) + f2 f 2 1 = 0 ∂z1 ∂z1 ∂z1 ∂z 2 i.e. ∂f2 ∂f ∂f
(f 1 f1 + f2 f 2 ) (f 1 − f1 ) 1 − f 2 − f2 1 = 0 ∂z1 ∂z1 ∂z 2 f 1 f2 f 2

∂f1 ∂f ∂f2 ∂f ∂f ∂f
+ f1 (f 1 − f1 ) 1 − f1 f 2 + f2 2 (f 1 − f1 ) − f1 f2 1 − f2 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂f1 ∂f ∂f2 ∂f ∂f ∂f
+ f 1f 2 + (f1 − f 1 )f 2 1 + f 2 f 2 − f 1 2 (f1 − f 1 ) + f2 f 2 1 − f 1 f2 2 = 0 ∂z1 ∂z1 ∂z1 ∂z1 ∂z 2 ∂z 2 By sum ∂f1 ∂f ∂f
f 2 f2 f 2 + f2 2 (f 1 − f1 ) − f2 f2 2 ∂z1 ∂z1 ∂z 2 ∂f1 ∂f ∂f
+f1 f 1 f 2 − f 1 2 (f1 − f 1 ) − f 1 f2 2 = 0 ∂z1 ∂z1 ∂z 2

f 2 f2 f 2 f1

i.e. f2

∂f1 ∂f 2 ∂f + (f − f1 ) − f2 2 = 0 ∂z1 ∂z1 1 ∂z 2

3.3. Definition. We will call w-hypermeromorphic function (w- for weak) any almost everywhere defined hyperholomorphic function whose right inverse is hyperholomorphic almost everywhere.

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4. On the spaces of hypermeromorphic functions. 4.1. Sum of two w-hypermeromorphic functions. Proposition 4.1. If f and g are two w-hypermeromorphic functions, then the following conditions are equivalent: (i) the sum h = f + g is w-hypermeromorphic; (ii) h satisfies the following PDE: −

∂|h| ∂ ∂|h|
∂
(h1 − h2 j) + |h| (h1 − h2 j) = 0 +j +j ∂z 1 ∂z 2 ∂z 1 ∂z 2

Proof. Explicit the condition: 2

|h| D(h−1 ) = −D(|h)|)(h) + |h|D(h) = 0; with h = h1 − h2 j

2Dh =

∂ ∂
∂h1 ∂h2 ∂h2 ∂h1
(h1 − h2 j) = j +j + − − ∂z 1 ∂z 2 ∂z 1 ∂z2 ∂z 1 ∂z2

1 ∂ ∂
(h1 h1 + h2 h2 ) +j 2 ∂z 1 ∂z 2 1 ∂h1 ∂h2 ∂h1 ∂h2
= h1 + h2 + h1 + h2 2 ∂z 1 ∂z 1 ∂z 1 ∂z 1 1 ∂h1 ∂h2 ∂h1 ∂h2
j = 0. + h1 + h2 + h1 + h2 2 ∂z2 ∂z2 ∂z2 ∂z2

D(|h|) = D(h1 h1 + h2 h2 ) =

4.2. Product of two w-hypermeromorphic functions. Proposition 4.2. Let f , g be two w-hypermeromorphic functions on U , then the following conditions are equivalent: (i) the product f ∗ g is w-hypermeremorphic; (ii) f and g satisfy the system of PDE: g1 (

∂f1 ∂f 2 ∂g1 ∂g1 ∂g + ) + (f1 − f 1 ) + f2 − f2 2 = 0 ∂z 1 ∂z2 ∂z 1 ∂z2 ∂z 1

g1 (

∂f1 ∂f 2 ∂g1 ∂g1 ∂g − ) + (f1 − f 1 ) − f2 − f2 2 = 0 ∂z 2 ∂z1 ∂z 2 ∂z1 ∂z 2

Proof. Let f = f1 + f2 j and g = g1 + g2 j two hypermeromorphic functions and f ∗ g =

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f1 g1 − f2 g 2 + (f1 g2 − f2 g 1 )j their product, then ∂f 2 ∂f1 − = 0; ∂z 1 ∂z2 ∂(f1 g1 − f2 g 2 ) ∂(f 1 g 2 − f 2 g1 ) − ∂z 1 ∂z2 ∂f1 ∂f ∂g ∂g1 ∂f 2 ∂f2 ∂g1 ∂g = g1 ( + ) − g2 ( 1 + ) + f1 − f1 2 + f2 − f2 2 = 0 ∂z 1 ∂z2 ∂z2 ∂z 1 ∂z 1 ∂z2 ∂z2 ∂z 1 ∂g ∂g1 ∂f1 ∂f 2 ∂g1 ∂g g1 ( + ) + f1 − f1 2 + f2 − f2 2 = 0. ∂z 1 ∂z2 ∂z 1 ∂z2 ∂z2 ∂z 1 ∂(f1 g1 − f2 g 2 ) ∂(f 1 g 2 − f 2 g1 ) + ∂z 2 ∂z1 ∂f ∂g ∂g1 ∂f1 ∂f 2 ∂f2 ∂g1 ∂g = g1 ( − ) + g2 ( 1 − ) + f1 − f1 2 + f2 − f2 2 = 0 ∂z 2 ∂z1 ∂z1 ∂z 2 ∂z 2 ∂z1 ∂z1 ∂z 2 ∂f 2 ∂g ∂g1 ∂f1 ∂g1 ∂g − ) + f1 + f1 2 − f2 − f2 2 = 0 g1 ( ∂z 2 ∂z1 ∂z 2 ∂z1 ∂z1 ∂z 2 4.3. Definition. We will call hypermeromorphic the w-hypermeromorphic functions whose sum and product are w-hypermeromorphic. Their space is nonempty, since it contains the space of the meromorphic functions. 5. Globalisation. Hamilton 4-manifold. 5.1.. The hypermeromorphic functions on a relatively compact open set U of H play the part of the meromorphic functions on a relatively compact open set U of C. We will call pseudoholomorphic function on U , every hypermeromorphic function, without poles on U . We will call smooth hypermeromorphic function (sha function) on U , every hypermeromorphic function, without zeroes and poles on U . Lemma 5.1. The quotient of two pseudoholomorphic functions on U , with the same zeroes and the same orders, is a sha function on U . ∼ C2 . Let X 5.2. Manifolds. The sha functions have been defined on open sets of H = be a 4-dimensional manifold bearing an atlas A of charts (hj , Uj ) such as the transition functions hi,j : Ui ∩ Uj → H are sha functions. X = (X, A) will be called an A-manifold analogous for H of a Riemann surface for C. I also propose to call an A-manifold a Hamilton 4-manifold. 5.3. Sheaves of pseudoholomorphic, hypermeromorphic functions. 5.3.1. Functions on an A-manifold X = (X, A). A map f : X → H is called a pseudoholomorphic function on X, if it is continuous and satisfies the following condition: for every chart (h, U ) of X, (f |U )h−1 : h(U ) → H) is a pseudoholomorphic. In the same way, a map f : X → H is called a hypermeromorphic function on X, if it is continuous and satisfies the following condition:for every chart (h, U ) of X, (f |U )h−1 : h(U ) → H) is a hypermeromorphic.

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5.3.2. Examples of Hamilton 4-manifold. The identity map of H is: z1 + z2 j 7→ z1 + z2 j. Ex. 1: (idH , H) is the unique chart of the atlas defining H as an A-manifold. Proof. The identity map (idH is f1 = z1 , f2 = z2 is pseudoholomorphic. Ex.2: Every open set V of X bears an induced structure of Hamilton 4-manifold. Ex. 3: Hamilton hypersphere HP. In the space HxH \ {0}, consider the equivalence relation ρ1 Rρ2 : there exists λ ∈ H∗ = H \ 0 such that ρ2 = ρ1 λ (right multiplication by λ). The elements of HxH \ {0} are the pairs (q1 , q2 ) 6= (0, 0). Let
π : HxH \ {0} → HxH \ {0} /R denoted HP. (q1 , q2 ) 7→ class of (q1 , q2 ) So, HP is the set of the quaternionic lines from the origin of H2 . Consider the pairs (q1 , q2 ) ∈ H2 , with q2 6= 0 we have: π(q1 , q2 ) = π(q1 q2−1 , 1); let ζ = q1 q2−1 , q2 6= 0; in the same way, consider the pairs (q1 , q2 ) ∈ H2 , with q1 6= 0 we have: π(q1 , q2 ) = π(1, q2 q1−1 ); let ζ 0 = q2 q1−1 , q1 6= 0. The charts ζ, ζ 0 have for domains U , U 0 , two open sets of HP, respectively homeomorphic to H forming an atlas of HP. Remark that U covers the whole of HP except the point π(q1 , 0) denoted ∞, and that U 0 covers the whole of HP except the point π(0, q2 ) denoted 0. U 0 = HP \ {0}. Over U ∩ U 0 , we have: ζ.ζ 0 = 1, i.e. ζ 0 = ζ −1 and ζ = q1 q2−1 . 5.3.3.. Pseudoholomorphic map or morphism. Let X and Y be two Hamilton 4-manifolds, a map f : X → Y is said pseudoholomorphic if it is continuous and if, for every pair of pseudoholomorphic charts (h, U ), (k, V ) such that f (U ) ⊂ V , k(f |U )h−1 : h(U ) → k(V ) be pseudoholomorphic. 5.3.4. Sheaf of pseudoholomorphic functions. Let U, V be two open sets of X such that U ⊂ V , then, the restrictions to U of the pseudoholomorphic functions on V are pseudoholomorphic on U . So is defined the sheaf, denoted P, of (non commutative rings) of pseudoholomorphic functions on X. The pair (X,P) is a ringed space. In the same way, the sheaf of non commutative rings, denoted M, of hypermeromorphic functions is defined on X. 5.3.5. Hamiltonian Submanifolds. They are submanifolds whose function ring is pseudoholomorphic. We will implicitly use the following fact: If f is a pseudoholomorphic or hypermeromorphic function, the same is true for a + f , where a is any fixed quaternion. The following examples are complex analytic submanifolds. i) H. Let a be a fixed quaternion, then a + C ⊂ H is a complex line from a embedded in H. ii) HP. Complex projective line imbedded in HP. Let i : z1 7→ z1 + z2 j and j : CP 7→ HP C×C\0 → H × H \ {0} i×i↓ ↓

C × C \ 0 /R0 → H × H \ {0} R Let p ∈ HP be a fixed point. Then, p + CP 1 is a complex projective line (or Riemann sphere) from p, embedded in HP.

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iii) Let S be a compact Riemann surface contained in HP as a Hamiltonian submanifold. Then p + S is a compact Riemann surface from p, embedded in HP. 5.3.6. A family of complex submanifolds in a Hamilton 4-manifold. We now use the properties and notions of subsection 2.5.2. They are a + Aα,β and also for restrtictions to an open set U of H. On a Hamilton 4-manifold X with an atlas A and every domain of chart U as above, we obtain: Proposition 5.2. Let (X, P) be a Hamilton 4-manifold. There exist a family of complex analytic curves Cb,γ,δ , of X. For every U domain of coordinates in A let Aγ,δ . By gluing, we get a complex analytic curve in (X, P) from b ∈ X, and γ, δ are real parameters. Proof. Let b ∈ X; β, γ ∈ R be given, consider an atlas A whose domains of charts are either open sets U of X disjoint from Aβ,γ , or Vβ,γ = U ∪ (Aβ,γ ∩ U ) where Aβ,γ ∩ U is connected, not empty. The restrictions of the charts of A to the U ∪ (Aβ,γ ∩ U ) define an atlas of Cb,γ,δ as complex analytic subvariety of (X, P), in the following way: assume b ∈ Vβ,γ ∩ Aβ,γ ∩ U and consider the open sets analogous to Vβ,γ such that the various Vβ,γ be connected. Then the corresponding Aβ,γ ∩ U constitute a covering of the unique complex analytic curve Cb,γ,δ . 5.3.7.. Let C be a complex analytic curve embedded into X and an atlas A such that every chart of domain U meeting C satisfies: U ∩ C is connected Theorem 5.3. The set of complex analytic curves in X is the family Cb,γ,δ . 6. Hamilton 4-manifold of a hypermeromorphic function. 6.1. Analytic continuation along a path. [D90, p. 116] Let X be a Hamilton 4-manifold, γ : [0, 1] → X a continuous path from a to b, ϕ ∈ Pa a germ of pseudoholomorphic function at a. Let τ ∈ [0, 1] and ϕτ ∈ Pγ(τ ) , there exists an open neighborhood Uτ of γ(τ ) in X and τ a pseudoholomorphic function fτ ∈ P(Uτ ) such that ρU γ(τ ) fτ = ϕτ . γ being continuous, it exists an open neighborhood Wτ of τ in [0,1] such that γ(Wτ ) ⊂ Uτ . 6.2. Definition. A germ ψ ∈ Pb is said to be the analytic continuation of ϕ along γ if there exists a family (ϕt )t∈[0,1] such that: 1) ϕ0 = ϕ and ϕ1 = ψ. τ 2) for every τ ∈ [0, 1], for every t ∈ Wτ , we have: ρU γ(τ ) fτ = ϕτ Theorem 6.1. Identity theorem. Let X be a connected Hamilton 4-manifold and f1 , f2 : X → Y be two morphisms which coincide in the neighborhood of a point x0 ∈ X, then f1 , f2 coincide on X. Proof as for Riemann surfaces, [D90, ch. 5]. Theorem 6.2. Let X be a simply connected Hamilton 4-manifold, a ∈ X, ϕ ∈ Pa be a germ having an analytic continuation along every path from a. Then there exists a unique function f ∈ P(X) such that ρX a f = ϕ.

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(cf. [D90, ch. 5, 4.1.5]) Let p : Y → X be a morphism of two Hamilton 4-manifolds; p is locally bi-pseudoholomorphic, then it defines, for every y ∈ Y , an isomorphism py : Px,p(y) → PY,y ; this defines: p∗ = p∗y = (p∗y )−1 . 6.3. Definition. Let X be a Hamilton 4-manifold, a ∈ X, ϕ ∈ Pa . A quadruple (Y, p, f, b) is called an analytic continuation of ϕ if: (i) Y is a Hamilton 4-manifold, p : Y → X is a morphism; (ii) f is a pseudoholomorphic function on Y ; (iii) b ∈ p−1 (a) ⊂ Y ; p∗ (ρYb f ) = ϕ. An analytic continuation is said to be maximal if it is solution of the following universal map problem: for every analytic continuation (Z, q, g, c) of ϕ, there exists a fibered morphism F : Z → Y such that F (c) = b and F ∗ (f ) = g. Hence If (Y, p, f, b) is a maximal analytic continuation of ϕ, it is unique up to an isomorphism. Y is called the Hamilton 4-manifold of ϕ. Theorem 6.3. Let X be a Hamilton 4-manifold, a ∈ X, ϕ ∈ Pa . Then there exists a maximal analytic continuation of ϕ. 6.4. Remark. Then, we will say that the above function f is the unique maximal analytic continuation of the germ ϕ. Moreover, the above definitions and results of the section 2 are valid for the sheaf M of hypermeromorphic functions instead of the sheaf P. 6.5. Main result. Theorem 6.4. Let X be a Hamilton 4-manifold and P (T ) = T n + c1 T n−1 + . . . + cn ∈ M(X)[T ] be an irreducible polynomial of degree n. Then there exist a Hamilton 4-manifold Y , a ramified pseudoholomorphic covering (cf. [D90, ch. 5] for Riemann surfaces) with n leaves Π : Y → X and a hypermeromorphic function F ∈ M(Y ) such that (Π∗ P )(F ) = 0. F is the unique maximal analytic continuation of every hypermeromorphic germ ϕ of X such that P (ϕ) = 0; F is called the hyperalgebraic function defined by the polynomial P and Y is the Hamilton 4-manifold of F . Proof at the end of the section. 1) X is compact connected. 2) Every pseudoholomorphic function on X is constant. 3) Every hypermeromorphic function f on X different from ∞ is rational. 4) In case X = HP, in Theorem 6.4, cj is rational. Indeed, since cj is hypermeromorphic, from 3), it is rational. 6.6. Proof of Theorem 6.4. In the notations of Ex. 3, ζ is a local coordinate on X = HP. f has a finite set of poles p1 , . . . , pn . Assume that ∞ is not a pole of f , then p1 , . . . , pn ∈ H. Let hν the principal part of f at pν , then f − hν = aν , constant, from 2) and hν = −1 X Cνj (ζ − pjν ) is a hypermeromorphic function, where Cνj ∈ H. j=−kν

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6.6.1. Elementary symmetric functions. Let Π:Y →X be a nonramified pseudoholomorphic covering with n leaves, and f be a hypermeromorphic function on Y . Every point x ∈ X has an open neighborhood U such that n [ Π−1 (U ) = Vj where the Vj are disjoint and Π|Vj : Vj → U is bi-pseudoholomorphic, j=1

(j = 1, . . . , n); let ϕj : U → Vj the reverse (i.e. set inverse) of Π|Vj and fj = ϕ∗j f = f.ϕj . Then: Πnj=1 (T − fj ) = T n + c1 T n−1 + . . . + cn ; cj = (−1)j sj (f1 , . . . , fn ), where sj is the j-th elementary symmetric function in n variables. The cj are hypermeromorphic, locally defined, but glue together into c1 , . . . , cn ∈ M(X) and are called the elementary symmetric functions of f with respect to Π. 6.6.2. Remark. The elementary symmetric functions of a hypermeromorphic function on Y are still defined when the covering Π is ramified. 6.6.3.. Theorem 6.5. Let Π as in Theorem 6.4, with Y not necessarily connected, A ⊂ X be a discreet closed subset containing all the critical values of Π, and B = Π−1 (A). Let f be a pseudoholomorphic (resp. hypermeromorphic) function on Y \ B and c1 , . . . , cn ∈ H(X \ A)(resp.M(X \ A)) the elementary symmetric functions of f . Then the following two conditions are equivalent: (i) f has a pseudoholomorphic (resp. hypermeromorphic) extension to Y ; (ii) for every j = 1, . . . , n, cj has a pseudoholomorphic (resp. hypermeromorphic) extension to X. 6.6.4. Existence of Y in Theorem 6.4. Let ∆ ∈ M(X) be the discriminant of P (T ); P (T ) being irreducible, ∆ 6= 0: then there exists a discrete closed set A ⊂ X such that, for every x ∈ X 0 = X \ A, ∆(x) 6= 0, and all the functions cj are pseudoholomorphic. Let Y 0 = {ϕ ∈ Hx , x ∈ X 0 ; P (ϕ) = 0} ⊂ LH, etal space defined by the sheaf H, and 0 Π : Y 0 → X, (ϕ 7→ x). It can be shown that, for every x ∈ X 0 , there exists an open neighborhood U of x in X 0 and functions fj ∈ H(U ), j = 1, . . . , n, such that P (T )|U = Πnj=1 (T − fj ); then n [ Π0−1 (U ) = [U, fj ] where [U, fj ] = {fjy , y ∈ U } is an open set of LH and Π0 |[U, fj ] : j=1

[U, fj ] → U is a homeomorphism; Y 0 is a Hamilton 2-manifold non necessarily connected, and a pseudoholomorphic, non ramified covering of X 0 . It can be shown that Π0 can be extended into a ramified pseudoholomorphic covering Π : Y → X of X for which Y 0 = Π−1 (X 0 ). The cj are defined on the whole of X; from Theorem 6.5, f has an extension F ∈ M(X) such that

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Π∗ P (F ) = F n + (Π∗ c1 )F n−1 + . . . + Π∗ cn = 0. It is easy to prove the connectedness of Y and the unicity of F . This ends the proof of Theorem 6.4. 7. The Hamilton 4-manifold Y of F when X = HP. 7.1. Recall the main properties of Y . Y is of real dimension 4; Y is connected; Y is compact; Y is C ∞ ; let m be the number of the critical values of Π and qj these critical values; they define points of Y forming the 0-skeleton of a simplicial complex K carried by the manifold Y . K may be supposed to be C ∞ by parts. Cutting along the 3-faces of K defines a fundamental domain F D of the covering Π. F D is a 4-dim polytope in HP with an even number of 3-faces; gluying together the opposite 3-faces, we get a compact 4-dim polytope with homology of the Hamilton 4-manifold Y . 7.2. Homology of Y . H p (Y ; Z), for p = 0, . . . , 4 have to be evaluated, using the critical values qj , and the Poincar´e duality.

References [CSSS04] [CLSSS07]

[D90]

[D13] [F39] [M66] [R13]

F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, Progress in Math. Physics 39, Birkh¨ auser (2004). F. Colombo, E. Luna-Elizarrar´ as, I. Sabadini, M.V. Shapiro, D.C. Struppa, A new characterization of pseudoconvex domains in C2 , Comptes Rendus Math. Acad. Sci. Paris, 344 (2007), 677-680. P. Dolbeault, Analyse complexe, Collection Maˆıtrise de math´ematiques pures, Masson Paris 1990; open access digitalized version available at http://pierre.dolbeault.free.fr/Book/P Dolbeault Analyse Complexe.pdf P. Dolbeault, On quaternionic functions, arXiv:1301.1320. ¨ R. Fueter, Uber einen Hartogs’schen Satz, Comm. Math. Helv. 12 (1939), 75-80. B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press (1966). G. Roos, Personnal communication, March 2013.